The problem involves solving a system of linear equations using the substitution method. The system provided is:

1. $x - 5y = -3$
2. $4x - 3y = 5$

Step-by-Step Solution:

Step 1: Solve one equation for one variable

From equation (1):
$x - 5y = -3$
Solve for $x$:
$x = 5y - 3$

Step 2: Substitute $x$ into the second equation

Substitute $x = 5y - 3$ into equation (2):
$4x - 3y = 5$
$4(5y - 3) - 3y = 5$
Expand:
$20y - 12 - 3y = 5$
Combine like terms:
$17y - 12 = 5$
Solve for $y$:
$17y = 17$
$y = 1$

Step 3: Solve for $x$ using $y = 1$

Substitute $y = 1$ into $x = 5y - 3$:
$x = 5(1) - 3$
$x = 5 - 3$
$x = 2$

Final Answer:

$x = 2, \, y = 1$

Would you like a detailed explanation of the steps or additional examples?

Related Questions:

1. Can the substitution method be used for nonlinear systems?
2. How does the elimination method compare to substitution for solving systems?
3. What are real-world applications of solving systems of equations?
4. How can graphing validate the solution to this system?
5. What is the determinant of a system matrix, and how does it relate to solutions?

Tip:

Always check your solution by substituting $x$ and $y$ back into both original equations to ensure they satisfy the system.